Local stability analysis using simulations and sum-of-squares programming

Ufuk Topcu; Andrew Packard; Peter Seiler

doi:10.1016/j.automatica.2008.03.010

Local stability analysis using simulations and sum-of-squares programming

Ufuk Topcu, Andrew Packard, Peter Seiler

Aerospace Engineering and Mechanics

Research output: Contribution to journal › Article › peer-review

175 Scopus citations

Abstract

The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.

Original language	English (US)
Pages (from-to)	2669-2675
Number of pages	7
Journal	Automatica
Volume	44
Issue number	10
DOIs	https://doi.org/10.1016/j.automatica.2008.03.010
State	Published - Oct 2008

Bibliographical note

Funding Information:
This work was sponsored by the Air Force Office of Scientific Research, USAF, under grant/contract number FA9550-05-1-0266. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the AFOSR or the US Government.

Keywords

Local stability
Nonlinear dynamics
Region-of-attraction
Simulations
Sum-of-squares programming

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10.1016/j.automatica.2008.03.010

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title = "Local stability analysis using simulations and sum-of-squares programming",

abstract = "The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.",

keywords = "Local stability, Nonlinear dynamics, Region-of-attraction, Simulations, Sum-of-squares programming",

author = "Ufuk Topcu and Andrew Packard and Peter Seiler",

note = "Funding Information: This work was sponsored by the Air Force Office of Scientific Research, USAF, under grant/contract number FA9550-05-1-0266. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the AFOSR or the US Government. ",

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AU - Topcu, Ufuk

AU - Packard, Andrew

AU - Seiler, Peter

N1 - Funding Information: This work was sponsored by the Air Force Office of Scientific Research, USAF, under grant/contract number FA9550-05-1-0266. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the AFOSR or the US Government.

PY - 2008/10

Y1 - 2008/10

N2 - The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.

AB - The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.

KW - Local stability

KW - Nonlinear dynamics

KW - Region-of-attraction

KW - Simulations

KW - Sum-of-squares programming

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JO - Automatica

JF - Automatica

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ER -

Local stability analysis using simulations and sum-of-squares programming

Abstract

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Keywords

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